A non-perturbative renormalization group study of the stochastic Navier--Stokes equation

TL;DR

This study applies a non-perturbative renormalization group approach to the stochastic Navier--Stokes equation, revealing fixed points and energy spectra consistent with classical turbulence laws, including Kolmogorov's -5/3 law.

Contribution

It introduces a non-perturbative approximation for the RG flow of the stochastic Navier--Stokes equation, capturing fixed points and scaling laws beyond perturbative predictions.

Findings

Convergence to a unique fixed point in RG flow across dimensions.

Recovery of Kolmogorov's -5/3 law at /2 psilon.

Identification of saturation in eddy diffusivity scaling at /2 psilon.

Abstract

We study the renormalization group flow of the average action of the stochastic Navier--Stokes equation with power-law forcing. Using Galilean invariance we introduce a non-perturbative approximation adapted to the zero frequency sector of the theory in the parametric range of the H\"older exponent $4-2\,\varepsilon$ of the forcing where real-space local interactions are relevant. In any spatial dimension $d$, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for $\varepsilon=2$ as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a \emph{saturation} in the $\varepsilon$-dependence of the scaling dimension of the eddy diffusivity at $\varepsilon=3/2$ when, according to perturbative renormalization, the velocity field becomes infra-red relevant.